Random Variable

Properties#

• Notation: the random variable is uppercase ($X$), a specific value lowercase ($x \in \mathbb{R}$). The MSI script follows this only up to §2.3; from §2.4 on, the lowercase letter denotes both, and densities are written $p(x)$ instead of $p_X(x)$.
• Rules for events:
  • the event containing all possible outcomes has probability $1$;
  • $A$, $B$ mutually exclusive (never true at the same time) ⇒ $P(A \vee B) = P(A) + P(B)$;
  • $A$, $B$ independent ⇔ $P(A \wedge B) = P(A)P(B)$. The joint probability $P(A \wedge B)$ is also written $P(A, B)$.
• Description by a density: a real-valued random variable is described by its Probability Density Function $p_X$; a random vector $X \in \mathbb{R}^n$ by a joint density on $\mathbb{R}^n$. Two scalar random variables $Y, Z$ can be stacked into the random vector $X = [Y, Z]^\top$ with $p_X(x) = p_{Y,Z}(y, z)$.
• Functions of random variables are random variables: $f(X)$ is itself a random variable, with Expected Value $\mathbb{E}\{f(X)\}$. In particular every estimate $\hat\theta(y)$ computed from random measurements $y$ is a random variable (Estimator).

Example#

Sources#

• MSI script p. 13 (§2.1), MSI Slides 02 > 2.1 Random Variables and Probabilities (not published)
• MSI script p. 16 (§2.3, random vectors), p. 17 (§2.4, notation change)